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Operatopes, Operanoids, and Noncommutative Zonoids

Eliza O'Reilly, Venkat Chandrasekaran

Abstract

We study a class of convex bodies called operatopes that are obtained by taking Minkowski sums of affine images of an operator norm ball. This notion generalizes that of zonotopes which are Minkowksi sums of line segments. Taking the limit of the number of line segments to infinity yields the class of convex bodies called zonoids, which can also be viewed as the expectation of a random line segment. Expanding on this interpretation, we analogously define operanoids as the expectation of a random affine image of an operator norm ball. In studying the properties of operanoids when the dimension of the operator norm ball grows, we arrive at a new asymptotic regime for limits of convex bodies. This leads to the more general class of convex bodies called noncommutative zonoids, and we use the framework of free probability theory to illustrate basic properties and examples. Finally, we discuss applications of operanoids and noncommutative zonoids in statistics and stochastic processes.

Operatopes, Operanoids, and Noncommutative Zonoids

Abstract

We study a class of convex bodies called operatopes that are obtained by taking Minkowski sums of affine images of an operator norm ball. This notion generalizes that of zonotopes which are Minkowksi sums of line segments. Taking the limit of the number of line segments to infinity yields the class of convex bodies called zonoids, which can also be viewed as the expectation of a random line segment. Expanding on this interpretation, we analogously define operanoids as the expectation of a random affine image of an operator norm ball. In studying the properties of operanoids when the dimension of the operator norm ball grows, we arrive at a new asymptotic regime for limits of convex bodies. This leads to the more general class of convex bodies called noncommutative zonoids, and we use the framework of free probability theory to illustrate basic properties and examples. Finally, we discuss applications of operanoids and noncommutative zonoids in statistics and stochastic processes.
Paper Structure (36 sections, 11 theorems, 90 equations, 6 figures)

This paper contains 36 sections, 11 theorems, 90 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Our Contributions
  3. Related Work
  4. Notation and Terminology
  5. Operatopes and Operanoids
  6. Definition
  7. More Examples
  8. Wigner random matrices
  9. Uniform permutation matrices
  10. Wishart random matrices
  11. Properties of Operanoids
  12. Facial Structure
  13. Relation between Operanoids and Other Families of Convex Bodies
  14. Operanoids in $\mathbb{R}^2$
  15. Operanoids and zonoids in higher dimensions
  16. ...and 21 more sections

Key Result

Theorem 2

A centrally symmetric convex body $Z$ in $\mathbb{R}^d$ centered at the origin is an $m$-operanoid if and only if the support function of $Z$ is of the form: for some $d$-tuple of random matrices $A_1, \ldots, A_d$ in $\mathbb{H}^m$ with $\mathbb{E}\left[\left(\sum_{i=1}^d \|A_i\|_2^2\right)^{\frac{1}{2}}\right] < \infty$.

Figures (6)

  • Figure 1: $3$-operatopes in $\mathbb{R}^2$ with one summand and the parametrizing matrices drawn from the GOE ensemble.
  • Figure 2: Samples of $6, 12$ and $20$-operatopes specified by i.i.d. uniformly random permutation matrices with $N = 150$ summands.
  • Figure 3: Samples of $\kappa m$-operatopes with $N = 150$ summands associated to i.i.d. Wishart matrices with Gaussian entries where $m = 6$ and $\kappa = 1/2, 1, 2$ from left to right.
  • Figure 4: A 3-dimensional 2-operatope that is not a zonoid.
  • Figure 5: Approximation of the limiting NC zonoids in $\mathbb{R}^2$ associated to free Haar unitaries (left) and to free Marchenko-Pastur random variables (right). These are each a random sample of an $m$-operatope with $N = 1$ summand and with $m=500$ associated to independent uniform permutation matrices and to independent Wishart random matrices, respectively.
  • ...and 1 more figures

Theorems & Definitions (38)

  • Definition 1
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Lemma 4
  • Proposition 5
  • proof
  • Theorem 6: Theorem 1.0.3 in HEINAVAARA2024
  • Proposition 7
  • ...and 28 more